1. Do some research on al-Khorezmi (also al-Khwarizmi), the man from whose name the word "algorithm" is derived. In particular, you should learn what the origins of the words "algorithm" and "algebra" have in common. 2. Given that the official purpose of the U.S. patent system is the promotion of the "useful arts," do you think algorithms are patentable in this country? Should they be? 3. a. Write down driving directions for going from your school to your home with the precision required from an algorithm's description. b. Write down a recipe for cooking your favorite dish with the precision required by an algorithm. 4. Design an algorithm for computing [vnfor any positive integer n. Besides assignment and comparison, your algorithm may only use the four basic arithmetical operations. 5. Design an algorithm to find all the common elements in two sorted lists of numbers. For example, for the lists 2, 5, 5, 5 and 2, 2, 3, 5, 5, 7, the output should be 2,5,5. What is the maximum number of comparisons your algorithm makes if the lengths of the two given lists are m and n, respectively? 6. a. Find ged(31415, 14142) by applying Euclid's algorithm. b. Estimate how many times faster it will be to find ged(31415, 14142) by Euclid's algorithm compared with the algorithm based on checking con- secutive integers from min{m, n) down to ged(m, n). 7. Prove the equality ged(m, n) = gedin, m mod n) for every pair of positive integers m and n. 8. What does Euclid's algorithm do for a pair of integers in which the first is smaller than the second? What is the maximum number of times this can happen during the algorithm's execution on such an input? 9. a. What is the minimum number of divisions made by Euclid's algorithm among all inputs 1